Finite Group Databasebeta

← #26243766

Subgroups — #26243766

Order 12032.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689163C211yesmaximal, characteristic, complemented
2C2×#15689128C211yesmaximal, characteristic, complemented
3C2×#15689128C211yesmaximal, characteristic, complemented
4C2×#15689087C211yesmaximal, characteristic, complemented
5#20245771C211yesmaximal, characteristic, complemented
6#20244574C214yesmaximal, complemented
7#20244567C214yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C22×#121×C47C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14#15689163—24no—
15C2×#13093957C2212yescomplemented
16C2×#13093955C2212yescomplemented
17#15689128—24no—
18#15689128—24no—
19C2×#13093945C2212yescomplemented
20C2×#13093945C2214yescomplemented
21C2×#13093945C2212yescomplemented
22#15689087—24no—
23C2×C22⋊(C47⋊C8)C2214yescomplemented
24#15688986—21no—
25#15688983C2212yescomplemented
26#15688983C2212yescomplemented
27#15688979—21no—
28C4×C22⋊Dic47—22no—
29#15688885C2214yescomplemented
30C23×D4×C47—21no—
31C23×D4×C47C2211yescharacteristic
32C24×C4×C47C2211yescharacteristic
33C2×#121×C47—24no—
34C2×#121×C47—22no—
35C2×#121×C47—24no—
36C2×#121×C47C2214yes—
37C22×C4⋊C4×C47C2211yescharacteristic
38C22×C22⋊C4×C47—21no—
39C2×#10625329—22no—
40C2×#10625329—22no—
41C2×#10625329—22no—
42C2×#10625329—22no—
43C2×#10625329—22no—
44C2×#10625329D414yescomplemented
45C22×C47⋊C8C2312yes—
46C22×C47⋊C8C2312yes—
47#13093983—22no—
48#13093983—22no—
49#13093983—22no—
50#13093983—22no—
51C2×C4⋊Dic47C2312yes—
52C2×C4⋊Dic47C2311yescharacteristic
53C2×C4⋊Dic47C2311yescharacteristic
54C2×C22⋊Dic47—22no—
55C2×C22⋊Dic47—22no—
56C2×C22⋊Dic47—21no—
57C2×C22⋊Dic47—21no—
58C2×C4×Dic47—21no—
59C2×C4×Dic47—21no—
60#13093957—24no—
61#13093955—24no—
62#13093945—24no—
63#13093945—24no—
64#13093945—28no—
65C22⋊(C47⋊C8)—28no—
66C47⋊#95—22no—
67C47⋊#95—22no—
68C25×C47—21no—
69C25×C47D411yescharacteristic
70C22×D4×C47—28no—
71C22×D4×C47—24no—
72C22×D4×C47—22no—
73C22×D4×C47—21no—
74C22×D4×C47—22no—
75C22×D4×C47—22no—
76C22×D4×C47—22no—
77C22×D4×C47—22no—
78C22×D4×C47—48no—
79C22×D4×C47D414yes—
80C22×D4×C47C2312yes—
81C23×C4×C47—21no—
82C23×C4×C47—22no—
83C23×C4×C47—22no—
84C23×C4×C47—22no—
85C23×C4×C47—22no—
86C23×C4×C47—22no—
87C23×C4×C47—22no—
88C23×C4×C47D412yes—
89C23×C4×C47D411yescharacteristic
90C23×C4×C47C2312yes—
91C23×C4×C47C2311yescharacteristic
92#121×C47—216no—
93#121×C47—28no—
94#121×C47—28no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47D414yes—
98C2×C4⋊C4×C47C2312yes—
99C2×C22⋊C4×C47—28no—
100C2×C22⋊C4×C47—24no—
101C24×C47—22no—
102C24×C47—24no—
103C24×C47—24no—
104C24×C47—22no—
105C24×C47—22no—
106C24×C47—22no—
107C24×C47—42no—
108C24×C47—44no—
109C24×C47—44no—
110C24×C47—42no—
111C24×C47—42no—
112C24×C47C2×D412yes—
113C24×C47C2×D411yescharacteristic
114C2×D4×C47—216no—
115C2×D4×C47—24no—
116C2×D4×C47—28no—
117C2×D4×C47—28no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47—22no—
121C2×D4×C47—22no—
122C2×D4×C47—22no—
123C2×D4×C47—432no—
124C2×D4×C47—48no—
125C2×D4×C47—48no—
126C2×D4×C47—44no—
127C2×D4×C47—42no—
128C2×D4×C47—42no—
129C2×D4×C47—44no—
130C2×D4×C47C2×D414yes—
131C22×C4×C47—22no—
132C22×C4×C47—24no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—24no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—22no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—44no—
146C22×C4×C47—44no—
147C22×C4×C47—44no—
148C22×C4×C47—44no—
149C22×C4×C47—44no—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D412yes—
153C22×C4×C47C2×D412yes—
154C22×C4×C47C2×D412yes—
155C22×C4×C47#4112yes—
156C22×C4×C47#4112yes—
157C22×C4×C47#4112yes—
158C22×C4×C47C2411yescharacteristic
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—22no—
162C4⋊C4×C47—24no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47C2×D414yes—
165C22⋊C4×C47—216no—
166C22⋊Dic47—44no—
167C22⋊Dic47—42no—
168C22⋊Dic47—42no—
169C22⋊Dic47—44no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—22no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—22no—
176C4×Dic47—42no—
177C4×Dic47—42no—
178#10625329—28no—
179#10625329—44no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8C2×D414yescomplemented
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—28no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—24no—
196C23×C47—48no—
197C23×C47—48no—
198C23×C47—44no—
199C23×C47—48no—
200C23×C47—48no—
201C23×C47—44no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—44no—
205C23×C47—44no—
206C23×C47—48no—
207C23×C47—88no—
208C23×C47—88no—
209C23×C47#12012yes—
210C23×C47#12112yes—
211C23×C47#12312yes—
212C23×C47AGL(1,ℤ/8)14yes—
213C23×C47C22×D411yescharacteristic
214D4×C47—24no—
215D4×C47—24no—
216D4×C47—432no—
217D4×C47—42no—
218D4×C47—42no—
219D4×C47—48no—
220C2×C4×C47—22no—
221C2×C4×C47—24no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—48no—
225C2×C4×C47—44no—
226C2×C4×C47—44no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—48no—
232C2×C4×C47#12012yes—
233C2×C4×C47#12114yes—
234C2×C4×C47#12114yes—
235C2×C4×C47#12112yes—
236C2×C4×C47#12314yes—
237C2×C4×C47#12312yes—
238C2×C4×C47#12414yes—
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C47⋊C8—24no—
253C47⋊C8—42no—
254#23480—471nomaximal, Sylow
255C22×C47—24no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—24no—
259C22×C47—24no—
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—416no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—44no—
268C22×C47—44no—
269C22×C47—44no—
270C22×C47—816no—
271C22×C47—84no—
272C22×C47—84no—
273C22×C47#39414yes—
274C22×C47C2×#12011yescharacteristic
275C22×C47C2×#12111yescharacteristic
276C22×C47C2×#12311yescharacteristic
277C22×C47C2×AGL(1,ℤ/8)12yes—
278C22×C47#57512yes—
279C4×C47—42no—
280C4×C47—44no—
281C4×C47#39414yes—
282C4×C47C2×#12111yescharacteristic
283C4×C47C2×#12111yescharacteristic
284C4×C47C2×#12311yescharacteristic
285C4×C47C2×#12411yescharacteristic
286Dic47—81no—
287Dic47—81no—
288Dic47—82no—
289C22×#121D4711yescharacteristic
290C2×#428—471no—
291C2×#418—471no—
292C2×#418—471no—
293C2×#407—471no—
294#2280—471no—
295#2034—474no—
296#2033—474no—
297#1736—471no—
298C2×C47—24no—
299C2×C47—44no—
300C2×C47—84no—
301C2×C47C2×#39411yescharacteristic
302C2×C47#300311yescharacteristic
303C2×C47#300511yescharacteristic
304C2×C47#305512yes—
305C2×C47#309412yes—
306C23×D4—21no—
307C23×D4C2×D4711yescharacteristic
308C24×C4C2×D4711yescharacteristic
309C22×M16—471no—
310C22×M16—471no—
311C2×#121—24no—
312C2×#121—22no—
313C2×#121—24no—
314C2×#121C2×D4714yes—
315C2×#117—471no—
316C2×#117—471no—
317C22×C4⋊C4C2×D4711yescharacteristic
318C22×C22⋊C4—21no—
319#428—944no—
320C2×#107—472no—
321C2×#106—472no—
322#418—944no—
323#418—944no—
324C2×#102—472no—
325C2×#102—474no—
326C2×#102—472no—
327#407—944no—
328C2×C22⋊C8—474no—
329#381—941no—
330#380—472no—
331#380—472no—
332#379—941no—
333C4×C22⋊C4—942no—
334#340—474no—
335C47#2348011yescharacteristic, complemented, Sylow
336C25—21no—
337C25C22⋊D4711yescharacteristic
338C22×D4—28no—
339C22×D4—24no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—22no—
346C22×D4—48no—
347C22×D4C22⋊D4714yes—
348C22×D4C22×D4712yes—
349C23×C4—21no—
350C23×C4—22no—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—22no—
356C23×C4C22⋊D4711yescharacteristic
357C23×C4C22⋊D4712yes—
358C23×C4C22×D4712yes—
359C23×C4C22×D4711yescharacteristic
360C2×M16—474no—
361C2×M16—942no—
362C2×M16—942no—
363C2×M16—942no—
364C2×M16—942no—
365C2×M16—942no—
366C22×C8—472no—
367C22×C8—472no—
368#121—216no—
369#121—28no—
370#121—28no—
371#117—942no—
372#117—942no—
373#117—942no—
374#117—942no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—472no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4C22⋊D4714yes—
381C2×C4⋊C4C22×D4712yes—
382C2×C22⋊C4—28no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4—942no—
387C2×C22⋊C4—942no—
388C2×C42—941no—
389C2×C42—941no—
390#107—944no—
391#106—944no—
392#102—948no—
393#102—944no—
394#102—944no—
395C22⋊C8—948no—
396#95—942no—
397#95—942no—
398C24—22no—
399C24—24no—
400C24—24no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—42no—
405C24—44no—
406C24—44no—
407C24—42no—
408C24—42no—
409C24C2×C22⋊D4712yes—
410C24C2×C22⋊D4711yescharacteristic
411C2×D4—216no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—432no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4C2×C22⋊D4714yes—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—941no—
448C22×C4—941no—
449C22×C4—942no—
450C22×C4D4×D4712yes—
451C22×C4#1062535212yes—
452C22×C4#1062535212yes—
453C22×C4#1062535212yes—
454C22×C4C2×C22⋊D4712yes—
455C22×C4C2×C22⋊D4711yescharacteristic
456C22×C4C2×C22⋊D4712yes—
457C22×C4C2×C22⋊D4712yes—
458C22×C4C23×D4711yescharacteristic
459M16—948no—
460M16—1884no—
461C2×C8—474no—
462C2×C8—942no—
463C2×C8—942no—
464C2×C8—942no—
465C2×C8—942no—
466C2×C8—942no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—942no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—942no—
475C4⋊C4C2×C22⋊D4714yes—
476C22⋊C4—216no—
477C22⋊C4—1884no—
478C22⋊C4—1882no—
479C22⋊C4—1882no—
480C22⋊C4—1884no—
481C42—1882no—
482C42—1882no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—48no—
497C23—48no—
498C23—44no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—88no—
505C23—88no—
506C23#1309399312yes—
507C23#1309399612yes—
508C23#1309400512yes—
509C23#1309405014yes—
510C23C22×C22⋊D4711yescharacteristic
511D4—24no—
512D4—24no—
513D4—432no—
514D4—48no—
515D4—42no—
516D4—42no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1882no—
536C2×C4#1309399312yes—
537C2×C4#1309399514yes—
538C2×C4#1309399612yes—
539C2×C4#1309399614yes—
540C2×C4#1309400512yes—
541C2×C4#1309400514yes—
542C2×C4#1309401114yes—
543C2×C4C2×D4×D4711yescharacteristic
544C2×C4C22×C22⋊D4711yescharacteristic
545C2×C4C22×C22⋊D4711yescharacteristic
546C2×C4C2×#1062535211yescharacteristic
547C2×C4C2×#1062535211yescharacteristic
548C2×C4C2×#1062535211yescharacteristic
549C8—944no—
550C8—1882no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—816no—
567C22—84no—
568C22—84no—
569C22#1568903714yes—
570C22C2×#1309399311yescharacteristic
571C22C2×#1309399611yescharacteristic
572C22C2×#1309400511yescharacteristic
573C22C2×#1309405012yes—
574C22#1568992112yes—
575C4—42no—
576C4—44no—
577C4—3762no—
578C4—3761no—
579C4—3761no—
580C4#1568903714yes—
581C4C2×#1309399511yescharacteristic
582C4C2×#1309399611yescharacteristic
583C4C2×#1309400511yescharacteristic
584C4C2×#1309401111yescharacteristic
585C2—24no—
586C2—44no—
587C2—84no—
588C2C2×#1568903711yescharacteristic
589C2#2025261511yescharacteristic
590C2#2025263011yescharacteristic
591C2#2025300312yes—
592C2#2025332412yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC22×#121×C47C2
socleC23×C47C22×D4
hypercenterC22×#121D47
hypocenterC47#23480
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#15689163C2semidirect
2. C2×#15689128C2semidirect
3. C2×#15689128C2semidirect
4. C2×#15689087C2semidirect
5. #20245771C2semidirect
6. #20244574C2semidirect
7. #20244567C2semidirect
8. #20243107C2semidirect
15. C2×#13093957C22semidirect
16. C2×#13093955C22semidirect
19. C2×#13093945C22semidirect
20. C2×#13093945C22semidirect
21. C2×#13093945C22semidirect
23. C2×C22⋊(C47⋊C8)C22semidirect
25. #15688983C22semidirect
26. #15688983C22semidirect
29. #15688885C22semidirect
44. C2×#10625329D4semidirect
185. C2×C47⋊C8C2×D4semidirect
335. C47#23480semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156891632
C2×#156891282
C2×#156890872
#202457712
#202445742
#202445672
#202431072
C22×#121×C472
#2348047